TY - JOUR
T1 - Quadrature Methods for Highly Oscillatory Singular Integrals
AU - Gao , Jing
AU - Condon , Marissa
AU - Iserles , Arieh
AU - Gilvey , Benjamin
AU - Trevelyan , Jon
JO - Journal of Computational Mathematics
VL - 2
SP - 227
EP - 260
PY - 2020
DA - 2020/11
SN - 39
DO - http://doi.org/10.4208/jcm.1911-m2019-0044
UR - https://global-sci.org/intro/article_detail/jcm/18373.html
KW - Numerical quadrature, Singular highly oscillatory integrals, Asymptotic analysis, Boundary Element Method, Plane wave enrichment, Partition of Unity.
AB - <p style="text-align: justify;">We address the evaluation of highly oscillatory integrals, with power-law and logarithmic singularities. Such problems arise in numerical methods in engineering. Notably, the
evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering, and many of these exhibit singularities. We show
that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the
integral. A truncated asymptotic expansion achieves an error that decays faster for increasing frequency. Based on the asymptotic analysis, a Filon-type method is constructed
to approximate the integral. Unlike an asymptotic expansion, the Filon method achieves
high accuracy for both small and large frequency. Complex-valued quadrature involves
interpolation at the zeros of polynomials orthogonal to a complex weight function. Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest
accuracy when the three methods are compared. However, while it achieves higher accuracy for the same number of function evaluations, it requires significant additional cost of
computation of orthogonal polynomials and their zeros.</p>